On super-rigidity of Gromov's random monster group

TL;DR

This paper proves that Gromov's random monster groups exhibit super-rigidity, meaning most morphisms into certain complex groups have finite images, revealing strong structural constraints of these random groups.

Contribution

It introduces the concept of hereditary super-rigidity and demonstrates that Gromov's random monster groups possess this property relative to specific classes of groups.

Findings

Most morphisms from the random monster group have finite images.

Gromov's random monster groups are hereditary super-rigid with respect to certain groups.

A stability theorem for super-rigidity properties is established.

Abstract

In this article, we show super-rigidity of Gromov's random monster group. We prove that any morphism $\phi_\alpha$ from Gromov's random monster group $\Gamma_\alpha$ to the group $G$ has finite image for almost all $\alpha$, where $G$ is any of the following types of groups: mapping class group $MCG(S_{g,b})$, braid group $B_n$, outer automorphism group of a free group $Out(F_N)$, automorphism group of a free group $Aut(F_N)$, hierarchically hyperbolic group, a-$L^p$-menable group or K-amenable group. We introduce another property called hereditary super-rigidity and prove that $\Gamma_\alpha$ has hereditary super-rigidity with respect to an a-$L^p$-menable group or a K-amenable group. We also establish a stability theorem for the groups with respect to which $\Gamma_\alpha$ has super-rigidity and hereditary super-rigidity.